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Now showing 1 - 10 of 14
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    Efficient and Robust Computation of an Approximated Medial Axis
    (The Eurographics Association, 2004) Yang, Y.; Brock, O.; Moll, R. N.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    The medial axis can be viewed as a compact representation for an arbitrary model; it is an essential geometric structure in many applications. A number of practical algorithms for its computation have been aimed at speeding up its computation and at addressing its instabilities. In this paper we propose a new algorithm to compute the medial axis with arbitrary precision. It exhibits several desirable properties not previously combined in a practical and ef cient algorithm. First, it allows for a tradeoff between computation time and accuracy, making it well-suited for applications in which an approximation of the medial axis suf ces, but computational ef ciency is of particular concern. Second, it is output sensitive: the computation complexity of the algorithm does not depend on the size of the representation of a model, but on the size of the representation of the resulting medial axis. Third, the densities of the approximated medial axis points in different areas are adaptive to local free space volumes, based on the assumption that a coarser approximation in wide open area can still suf ce the requirements of the applications. We present theoretical results, bounding the error introduced by the approximation process. The algorithm has been implemented and experimental results are presented that illustrate its computational ef ciency and robustness.
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    Automatic Building of Structured Geological Models
    (The Eurographics Association, 2004) Brandel, S.; Schneider, S.; Perrin, M.; Guiard, N.; Rainaud, J. F.; Lienhardt, P.; Bertrand, Y.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    The present article proposes a method to signi cantly improve the construction and updating of 3D geological models used for oil and gas exploration. The proposed method takes advantage of the speci c structures which characterize geological objects. We present a prototype of a geological pilot which enables monitoring the automatic building of a 3D model topologically and geologically consistent, starting from a set of unsegmented surfaces. The geological pilot uses a Geological Evolution Scheme (GES) which records all the interpretation elements that the exploration geologist, who is the end user, wishes to introduce into the model. The model building is performed by reading instructions deduced from the GES. Topology is dealt with step by step by using a 3D Generalized Maps (3-G-Maps) data model enriched to enable the manipulation of objects having speci c geological attributes. The result is a correct 3D model on which geological links between objects can easily be visualized. This model can automatically be revised in case of changes in the geometric data or in the interpretation. In its nal version, the created modular tool will be plugged in 3D modelers currently used in exploration geology in order to improve their performance.
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    Update Operations on 3D Simplicial Decompositions of Non-manifold Objects
    (The Eurographics Association, 2004) Floriani, L. De; Hui, A.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    We address the problem of updating non-manifold mixed-dimensional objects, described by three-dimensional simplicial complexes embedded in 3D Euclidean space. We consider two local update operations, edge collapse and vertex split, which are the most common operations performed for simplifying a simplicial complex. We examine the effect of such operations on a 3D simplicial complex, and we describe algorithms for edge collapse and vertex split on a compact representation of a 3D simplicial complex, that we call the Non-Manifold Indexed data structure with Adjacencies (NMIA). We also discuss how to encode the information needed for performing a vertex split and an edge collapse on a 3D simplicial complex. The encoding of such information together with the algorithms for updating the NMIA data structure form the basis for de ning progressive as well as multi-resolution representations for objects described by 3D simplicial complexes and for extracting variable-resolution object descriptions.
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    Hardware-based Simulation and Collision Detection for Large Particle Systems
    (The Eurographics Association, 2004) Kolb, A.; Latta, L.; Rezk-Salama, C.; Tomas Akenine-Moeller and Michael McCool
    Particle systems have long been recognized as an essential building block for detail-rich and lively visual environments. Current implementations can handle up to 10,000 particles in real-time simulations and are mostly limited by the transfer of particle data from the main processor to the graphics hardware (GPU) for rendering. This paper introduces a full GPU implementation using fragment shaders of both the simulation and rendering of a dynamically-growing particle system. Such an implementation can render up to 1 million particles in real-time on recent hardware. The massively parallel simulation handles collision detection and reaction of particles with objects for arbitrary shape. The collision detection is based on depth maps that represent the outer shape of an object. The depth maps store distance values and normal vectors for collision reaction. Using a special texturebased indexing technique to represent normal vectors, standard 8-bit textures can be used to describe the complete depth map data. Alternately, several depth maps can be stored in one floating point texture. In addition, a GPU-based parallel sorting algorithm is introduced that can be used to perform a depth sorting of the particles for correct alpha blending.
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    Optimization Techniques for Approximation with Subdivision Surfaces
    (The Eurographics Association, 2004) Marinov, M.; Kobbelt, L.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    We present a method for scattered data approximation with subdivision surfaces which actually uses the true representation of the limit surface as a linear combination of smooth basis functions associated with the control vertices. This is unlike previous techniques which used only piecewise linear approximations of the limit surface. By this we can assign arbitrary parameterizations to the given sample points, including those generated by parameter correction. We present a robust and fast algorithm for exact closest point search on Loop surfaces by combining Newton iteration and non-linear minimization. Based on this we perform unconditionally convergent parameter correction to optimize the approximation with respect to the L2 metric and thus we make a well-established scattered data tting technique which has been available before only for B-spline surfaces, applicable to subdivision surfaces. Further we exploit the fact that the control mesh of a subdivision surface can have arbitrary connectivity to reduce the L1 error up to a certain user-de ned tolerance by adaptively restructuring the control mesh. By employing iterative least squares solvers, we achieve acceptable running times even for large amounts of data and we obtain high quality approximations by surfaces with relatively low control mesh complexity compared to the number of sample points. Since we are using plain subdivision surfaces, there is no need for multiresolution detail coef cients and we do not have to deal with the additional overhead in data and computational complexity associated with them.
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    An Effective Condition for Sampling Surfaces with Guarantees
    (The Eurographics Association, 2004) Boissonnat, J. D.; Oudot, S.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    The notion of e-sample, as introduced by Amenta and Bern, has proven to be a key concept in the theory of sampled surfaces. Of particular interest is the fact that, if E is an e-sample of a smooth surface S for a suf ciently small e, then the Delaunay triangulation of E restricted to S is a good approximation of S, both in a topological and in a geometric sense. Hence, if one can construct an e-sample, one also gets a good approximation of the surface. Moreover, correct reconstruction is ensured by various algorithms. In this paper, we introduce the notion of loose e-sample. We show that the set of loose e-samples contains and is asymptotically identical to the set of e-samples. The main advantage of loose e-samples over e-samples is that they are easier to check and to construct. We also present a simple algorithm that constructs provably good surface samples and meshes.
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    A Framework for Multiresolution Adaptive Solid Objects
    (The Eurographics Association, 2004) Chang, Y.- S.; Qin, H.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    Despite the growing interest in subdivision surfaces within the computer graphics and geometric processing communities, subdivision approaches have been receiving much less attention in solid modeling. This paper presents a powerful new framework for a subdivision scheme that is defined over a simplicial complex in any n-D space. We first present a series of definitions to facilitate topological inquiries during the subdivision process. The scheme is derived from the double (k+1)-directional box splines over k-simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a regular mesh. The subdivision rules are modified by spatial averaging to guarantee C1 smoothness near extraordinary cases. Within a single framework, we combine the subdivision rules that can produce 1-, 2-, and 3-manifold in arbitrary n-D space. Possible solutions for non-manifold regions between the manifolds with different dimensions are suggested as a form of selective subdivision rules according to user preference. We briefly describe the subdivision matrix analysis to ensure a reasonable smoothness across extraordinary topologies, and empirical results support our assumption. In addition, through modifications, we show that the scheme can easily represent objects with singularities, such as cusps, creases, or corners. We further develop local adaptive refinement rules that can achieve level-of-detail control for hierarchical modeling. Our implementation is based on the topological properties of a simplicial domain. Therefore, it is flexible and extendable. We also develop a solid modeling system founded on our theoretical framework to show potential benefits of our work in industrial design, geometric processing, and other applications.
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    Constraint-based Design of B-spline Surfaces from Curves
    (The Eurographics Association, 2004) Michalik, P.; Bruderlin, B. D.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    In this paper we describe the design of B-spline surface models by means of curves and tangency conditions. The intended application is the conceptual constraint-driven design of surfaces from hand-sketched curves. The solving of generalized curve surface constraints means to find the control points of the surface from one or several curves, incident on the surface, and possibly additional tangency and smoothness conditions. This is accomplished by solving large, and generally under-constrained, and badly conditioned linear systems of equations. For this class of linear systems, no unique solution exists and straight forward methods such as Gaussian elimination, QR-decomposition, or even blindly applied Singular Value Decomposition (SVD) will fail. We propose to use regularization approaches, based on the so-called L-curve. The L-curve, which can be seen as a numerical high frequency filter, helps to determine the regularization parameter such that a numerically stable solution is obtained. Additional smoothness conditions are defined for the surface to filter out aliasing artifacts, which are due to the discrete structure of the piece-wise polynomial structure of the B-spline surface. This leads to a constrained optimization problem, which is solved by Modified Truncated SVD: a L-curve based regularization algorithm which takes into account a user defined smoothing constraint.
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    Contour Interpolation with Bounded Dihedral Angles
    (The Eurographics Association, 2004) Bereg, S.; Jiang, M.; Zhu, B.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids generated by any contour interpolation method. Given two arbitrary parallel contour slices with n vertices in 3D, let a be the smallest angle in the constrained Delaunay triangulation of the corresponding 2D contour overlay, we present a contour interpolation method which reconstructs a 3D solid with the minimum dihedral angle of at least a 8 . Our algorithm runs in O(nlogn) time where n is the size of the contour overlay. We also present a heuristic algorithm that optimizes the dihedral angles of a mesh representing a surface in 3D.
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    Multiresolution Heterogeneous Solid Modeling and Visualization Using Trivariate Simplex Splines
    (The Eurographics Association, 2004) Hua, J.; He, Y.; Qin, H.; Gershon Elber and Nicholas Patrikalakis and Pere Brunet
    This paper presents a new and powerful heterogeneous solid modeling paradigm for representing, modeling, and rendering of multi-dimensional, physical attributes across any volumetric objects. The modeled solid can be of complicated geometry and arbitrary topology. It is formulated using a trivariate simplex spline defined over a tetrahedral decomposition of any 3D domain. Heterogeneous material attributes associated with solid geometry can be modeled and edited by manipulating the control vectors and/or associated knots of trivariate simplex splines easily. The multiresolution capability is achieved by interactively subdividing any regions of interest and allocating more knots and control vectors accordingly. We also develop a feature-sensitive fitting algorithm that can reconstruct a more compact, continuous trivariate simplex spline from structured or unstructured volumetric grids. This multiresolution representation results from the adaptive and progressive tetrahedralization of the 3D domain. In addition, based on the simplex spline theory, we derive several theoretical formula and propose a fast direct rendering algorithm for interactive data analysis and visualization of the simplex spline volumes. Our experiments demonstrate that the proposed paradigm augments the current modeling and visualization techniques with the new and unique advantages.